Translated from the original Japanese version, OYO BUTURI
Vol.72, No.2, p.146 (2003)

Single photons for quantum
information systems
Yoshihisa YAMAMOTOThis paper introduces the current stateof the development of single photon sourcesthat will play a central role in future quantum information systems. By optically pumping a system consisting of a semiconductor
single quantum dot confined in a monolithic
microcavity, it is possible to produce a single
photon pulse stream at the Fourier transform limit with high efficiency and with ahigh repetition speed. It is expected that thistechnique will not only prove to be usefulfor BB84 quantum cryptography using single
photons, but will also find applications in other fields such as BBM92 quantum cryptography using entangled photon pairs, quantum
teleportation, quantum repeaters, and linear
optical quantum computers.
Keywords: single photon source, quantum
dot, microcavity, Fourier transform limited, EPR-Bell state,
two photon interference, quantum cryptography, quantum
teleportation, quantum computation

1. Introduction
Improvements in the protection of privacy and the ability to analyze complex systems are important issues to be addressed in
the fields of information science and physical science in the 21st century. The former
is needed to guarantee the security of personal information in telecommunications andcomputer systems, and the latter is needed
to realize computer hardware and softwarethat can process very large amounts of datain fields such as predicting weather patterns,
analyzing ecological phenomena, controlling
traffic and estimating stock market fluctuations.
Quantum information research has pro-

gressed rapidly in recent years in the hope ofaddressing these needs. This field of researchcovers many topics, and is developing at an
unprecedented scale with the involvement ofresearchers from many different leading-edge
science and technology fields. Although it isvery hard to predict how things will turn out
in the future, it seems that as far as a hardware is concerned, important roles will beplayed by photonic qubits for quantum communication, nuclear spin qubits for quantum
memory, and electron spin qubits for quantum information processing to connect them
together. This paper introduces the latest research trends relating to the production ofphotonic qubits used for quantum communication and computation, with particular reference to the results obtained by the author’sgroup.
In quantum cryptography,1) the securityof schemes that involve communication using
an ordinary Poisson light source (such as theBB84 protocol which uses a semiconductor
laser) can be threatened by an eavesdropper’s
photon splitting attack (i.e., secretly extracting and measuring a single photon when
there are 2 or more photons in a pulse).2)
Even in the Ekert91 or BBM92 protocols forquantum cryptography using EPR-Bell photon
pairs produced by parametric downconversion,3) bit errors occur due to the presence of
two or more photon pairs per pulse.4) Hopes
are therefore pinned on the development of
a light source that can generate single photons or single EPR-Bell photon pairs at definite time intervals.
Methods have been proposed for implementing optical quantum information processing techniques that are more advanced
than quantum cryptography, such as quantum teleportation,5) quantum repeaters,6) andlinear optics quantum computers.7) For suchapplications, it is essential to generate large
numbers of single photons at definite timeintervals that satisfy the condition of “indistinguishable quantum particles”.
Various methods are currently being investigated for the production of single photons at definite time intervals, including
methods that use single atoms, single ions,
single molecules or single solid-state latticedefects (color centers). However, the resultsof these studies have yet to satisfy various
conditions such as high efficiency, high speed
and single-mode operation.
The author’s group has previously proposed and verified a method for producing
single photons regularly by using the Coulomb blockade effect in a micro-pn junction
with a quantum well as the active layer.8,9)
However, the following three problems madeit difficult to use this turnstile device in practical systems:(1)To realize the Coulomb blocking effect fora single electron and hole, the device had
to be operated at the extremely low temperature of 50 mK.
(2)The electron-hole pairs have a long radiative recombination lifetime of 25 ns, making it impossible to operate the device athigh repetition rates.
(3)The photons are radiated in random directions, resulting in a poor coupling efficiency to an external optical system.
A conceivable way of addressing these
problems is to generate single photons byusing excitonic emission from a single InAs/
GaAs quantum dot excited by a pulse oflight. The following experiments will be discussed in this paper: generating a stream of
single photons from a single quantum dot,10)
using a DBR post microcavity to increase theSingle photons for quantum information systems. Yoshihisa YAMAMOTO. Quantum Entanglement Project, ICORP, JST, Stanford University (Stanford, CA 94305-4088 U.S.A.)
NTT Basic Research Laboratories

4

JSAP International No.10 (July 2004)

No. of counts

4000

0.97

0.99

0.97

1.00

1.00

0.75

0.75

0.50

0.50

0.25

0.25

0.98

3000
2000
1000
0

-26

-13
0
Delay time τ =t2-t1 (ns)

13
0

0.25

0.50
T1

Fig. 1: Measurement results for a Poisson light source(pulse stream from a Ti:Al2O3 laser)

efficiency of the external coupling efficiency,11) generating a stream of single photons
whose spectral width and pulse width are at
the Fourier transform limit,12) experiments inwhich single photons at this Fourier limit are
regarded as indistinguishable single quantum
particles (two-photon interference phenomenon),13) and BB84 quantum key distribution
experiments using single photons generated
in this way.14)

2. Producing single photons
2.1 Photon –photon correlation
The second-order coherence function of
the optical field that characterizes the difference between a single photon source and an
ordinary laser light source is defined by thefollowing equation:g (2) (τ )=〈a (t)a (t+τ )a(t+ τ )a(t)〉〈a + a〉 2++

where a (a +) is the photon annihilation(creation) operator.
Figure 1 shows the results of measurements on an optical pulse stream from a
mode-locked Ti:sapphire laser. A Hanbury
Brown and Twiss optical intensity interferometer (described below) was used to make

0.75

0
1.00
µm

0

0.25

0.50

0.75

T2 >T10
1.00µm

Fig. 2: AFM image of self-assembled InAs/GaAs quantum dots made by MBE deposition

these measurements. The experimental results shown in Fig. 1 represent the histogramof photons detected at other times whena single photon was detected at time t = 0.
Since all the peaks including the peak at t = 0
are equal in height, this shows that in thepulses from a Ti:sapphire laser, when a single
photon is detected in the pulse that arrives
at time t = 0, the probability that a second
photon will be detected in the same pulse isequal to the probability that the first photon
will be detected in a pulse that arrives at another time. This is a unique characteristic of
optical pulse streams where the photon numbers have a Poisson distribution.
In the case of a super-Poisson light source
where the distribution of photon numbers is
wider than a Poisson distribution, the probability that a second photon will be detectedin the same pulse in which a single photon isdetected is greater than the probability thatthe first photon will be detected in anotherpulse. This phenomenon is called photon
bunching, and this characteristic is exhibitedby light emitted from a source in thermal
equilibrium. It originates from the stimulated
emission of bosonic particles. Conversely, inthe case of a sub-Poisson light source where

the distribution of photon numbers is narrower than a Poisson distribution, the probability that a second photon will be detectedin the same pulse in which a single photon isdetected is less than the probability that thefirst photon will be detected in another pulse.
This phenomenon is called photon antibunching. Since each pulse obtained from a single
photon source always consists of a single
photon, g (2)(τ ) is given by the following equation:g (2) (τ )={0 ( τ =0)
1 ( τ =nT )where T is the light pulse repetition interval, and n is a non-zero integer.

2.2 Emission spectrum froma single quantum dot
Figure 2 shows an AFM image of self-assembled InAs/GaAs quantum dots producedby molecular beam epitaxy (MBE). By setting a high deposition temperature, it is possible to reduce the surface density of quantum dots. By using a combination of electron
beam lithography and dry etching techniques
to process this wafer into posts with a diameter of 0.2 µ m, it is possible to optically excite

JSAP International No.10 (July 2004)

5

Streak camera

CCD

Spectrometer

Half-wavelength plate
Polarizer

Beam splitter
Photon counter
Start
Time difference
measurement
Stop

PinholeCryostat
GratingMode-lockedtitanium sapphire laser

10 nm band-pass filter

Fig. 3: Single quantum dot light emission measurement systemhigher energy than the GaAs band gap. The
emission patterns from different quantum
dots have basically the same pattern except
for relative shifts in wavelength. Emission
peaks 1 and 2 correspond to single-excitonand biexciton emissions from a neutral quantum dot, respectively. The emission wavelength difference results from multiple carrierinteractions, and correspond to the exciton
molecule binding energy. Emission peaks 3
and 4 indicate the exciton (trion) emission
from quantum dots that have trapped a holeand an electron respectively and have positive

(a)

2
4

1
3

882 880 878 876 874
Wavelength (nm)Intensity (arbitrary units)

Intensity (arbitrary units)

a single quantum dot in isolation. The experimental step is shown in Fig. 3. The temporal
response to light emission and the spectral
characteristics were measured with a streak
camera and a grating spectrometer. The Hanbury Brown and Twiss interferometer used tomeasure g (2)(τ ) consists of a 50-50% beam
splitter, an optical band-pass filter, a single
photon counting Si APD (SPCM), and a delay
time measuring circuit.
Figure 4(a) shows the emission spectrumof the first quantum level (1 e-1 h) in the casewhere optical pumping is performed at a

(b)

880

878 876 874
Wavelength (nm)100

Fig. 4: (a) Emission spectrum
from an InAs single quantum
dot excited at energies higher
than the GaAs band edge.
(b) Emission spectrum from an
InAs single quantum dot when
the second quantization levelis resonantly excited.
(c) Variation of emission intensity with CW pump power foremission lines 1 (앪) and 2 (앳).

6

Intensity (arbitrary units)

(c)

10-1
Peak 1

Peak 210-2
10-3

JSAP International No.10 (July 2004)

10-2

10-1

Pump power (mW)

100

and negative charges.
Figure 4(b) shows the emission spectrum from the first quantum energy level (1e-1 h) when an InAs quantum dot is excitedby resonant optical pumping to the second
quantum energy level (2 e-2 h). It is known
that resonant optical pumping can suppress
the emission peaks 3 and 4 from the quantum dot. This fact has important significancefor the realization of a single photon sourceat the Fourier transform limit described below.
Figure 4(c) shows that in the low pumping region the emission peaks 1 and 2 increase in proportion linearly and quadratically to the pumping intensity. The respective
emission dependences on the pump power
support the interpretation of exciton and biexciton emissions. These experimental results
relate to CW pumping, and when the steadystate average number of excitons in thequantum dot exceeds 1, the probability thata single exciton is trapped in the quantum
dot decreases and the strength of the exciton
emission peak is reduced.
Figure 5(a)-(d) shows the temporal response of the exciton, biexciton and triexciton emission intensities when pumped withlight pulses of various intensities.15) The double lines show the theoretical values basedon two assumptions: (i) that the number of
excitons injected into the quantum dot at
t = 0 follows the Poisson distribution of theaverage value µ , and (ii) that each exciton independently releases a photon with a fixedemission lifetime of τ . The experimental results can be well explained by selecting the

1

two parameters of the average number of
excitons µ and the photodetection quantum
efficiency I 0. When the number of excitons(a)µ =0.658
I0=3.169(b)µ =1.344
I0=1.506

0.5

injected into the quantum dot at t = 0 is threeor more, no triexciton emission occurs until
the number of excitons in the quantum dot
reaches 3. Also, biexciton emission does notoccur until the triexciton emission has finishedand the number of excitons in the quantum
dot has reached 2. The process leading up to
the exciton emission is the same. Figure 5(c)shows the results of measurements with astreak camera indicating this state.

0

Intensity (arbitrary units)

0.6
0.4
0.2

É

0
1

(c)

µ =2.818
I0=1.912(d)µ =5.532
I0=1.750

0.5

2.3 Post filtering
0
1

0.5

0

Time (ns)

(e)

0

27 mW

0.5

1
Time (ns)1.5

108 mW

432 mW

0

0

0

0.5

0.5

0.5

1

1

1

1.5

1.5

1.5

880 879 878 877 876
Wavelength (nm)880 879 878 877 876
Wavelength (nm)880 879 878 877 876
Wavelength (nm)Fig. 5: Delay characteristics of exciton emission, biexciton emission and triexciton emission with pump pulse intensitiesof (a) 27 µW, (b) 54 µW, (c) 108 µW, and (d) 432 µW. Here, µ is the average number of excitons at time t=0, and I0 is aparameter proportional to the detection efficiency. (e) Time delay vs. wavelength characteristics of light emission froma single quantum dot, as observed with a streak camera.

×10

No. of counts

3

8
7
6

A

B

C

D

4
3

E

P0=0.37 mW

1.59

2.06

0.44 mW
0.036

2.07

1.31

1.74

0.88 mW
0.039

1.64

1.11

1.47

1.32 mW
0.077

1.51

50

0

5
No. of counts

No. of counts

500I0=7444 cps

2

No. of counts

The last photon to be released from aquantum dot that initially has multiple excitons has a unique emission wavelength andis the only photon with this wavelength. By
using an optical band-pass filter to extract
just this last photon, it is possible to generate a single photon corresponding to eachpump pulse. Figure 6(a) shows how the post
filtered exciton emission intensity obtainedin this way varies with the intensity of thepumping light.10) The strongly saturated characteristics show that when one or more excitons are injected into the quantum dot, thelast photon to be generated is always extracted by the band-pass filter. Figure 6(b) shows
the results of measuring g (2)(τ ) from a streamof single photons generated in this way.10)
Here, the unique characteristics of a single
photon source – g (2)(0)≈0, and g (2)(τ = nT )= 1
– are more or less realized. The small residual
value of g (2)(0) is due to insufficient suppression of the optical filter’s stop bands. Close tot =0, the value of g (2)(τ = nT ) becomes greaterthan 1 because the light emission from thequantum dot is superimposed on the on-off
modulation (blinking effect). This results in a
repeating pattern whereby the quantum dot
emits light at the wavelength being measured
at a certain time, after which the light emission at this wavelength stops, and then restarts again. This is thought to be caused bya shift in exciton emission wavelength resulting from the capture and release of carriers
into and out from carrier traps in the vicinityof the quantum dot. Similar reports relating
to the generation of single photons using asingle quantum dot have been reported byseveral research groups.16-19)

1
0
0

1

2

3
Pump (mW)

(a)

4

5

6

50

0
-30

-20

-10
0
Delay time τ (ns)

10

(b)

Fig. 6: (a) Exciton emission intensity vs. optical pump power. (b) Results of measuring g(2)(τ ) of exciton emissions from asingle quantum dot at various different optical pump intensities.

JSAP International No.10 (July 2004)

7

3. Controlling spontaneousemission with a single-mode
cavityformed DBR micropost cavity, its behavior wasstudied by the first principle, i.e., using thefinite difference time domain method.20,21) A

Figure 7(a) shows an SEM photographof a three-dimensional microcavity produced
by using electron beam lithography and ECR
dry etching to process a DBR planar cavity made by MBE into a post shape. An InAs
quantum dot is embedded in the central optical cavity layer.
Figure 8 shows the emission spectrumof a DBR microcavity with a post diameter of
6 µ m.11) In a post system of this size, large
numbers of InAs quantum dots having nonuniform spreading contribute to the emission
spectrum. Since different transverse modes(HE11, HE21, ...) have different longitudinalwavenumber, they each have different resonant wavelengths. The arrows show the theoretical values of the resonant wavelengthfor each mode, and explain the experimental
results well.
Figure 9(a) and (b) show the results ofmeasuring the wavelength dependence ofthe emission spectrum and emission lifetimeof the HE11 fundamental mode of DBR microcavities with post diameters of 2 µ m and0.5 µ m.11) The fact that the emission lifetimeis shortest at the resonant wavelength of thecavity is a unique characteristic of a three-dimensional cavity. In general, the spontaneousemission rate at the resonant wavelength ofa three-dimensional cavity normalized by the
spontaneous emission rate γ 0 in free space iscalled the Purcell factor, as expressed by thefollowing formula:post with a diameter of 0.4 µ m and a heightof 5 µ m was predicted to achieve a Q valueof 10000, a mode volume V of 1.5 (λ /n)3, a
Purcell factor F of 100, and a spontaneousemission coefficient β of 0.99.20) This value
is at least an order of magnitude better thanthe value measured experimentally. This discrepancy is thought to occur for the follow-

HE11

1
β =1–
F
To investigate the Q value of an ideally

JSAP International No.10 (July 2004)

Fig. 7: GaAs/AlAs DBR micropost cavities
enclosed in the center of InAs quantum
dots produced by electron beam lithography and (a) Electron Cyclotron Resonance(ECR) dry etching, (b) Chemically Assisted
Ion Beam Etching (CAIBE).

(b)

EH01 HE21 HE01

Intensity (arbitrary units)

EH11...

∆λ 2c
Qλ 3
E 2
2
2 3
2π n V 0 E max ∆λ c +4(λ – λ c ) 2

HE12...
HE21...
HE02...
EH31...
EH12...

EH13

HE13...

HE52

HE03...

1.330

1.335

1.340

1.345

Energy (eV)Fig. 8: Resonant modes of a microcavity
with a post diameter of 6 µm(b) 0.6

(a)
1.5
1.2
0.9
0.6
929
930
928
Wavelength (nm)Emission intensity(arbitrary units)

Here, λ is the emission wavelength, λ c is
the resonant wavelength of the cavity, V 0 is
the cavity’s mode volume, E max is the maximum electric field inside the cavity, E is theelectric field at the position of the atom, and∆λ c /λ c Q is the resonant width of the cavity. The ratio of the rate of spontaneous emissions into the single cavity mode to the overall spontaneous emission rate is called thespontaneous emission coefficient, which isexpressed as follows:

8(a)Emission lifetime τ (ns)0cavity’s Q value increases, as the mode volume V 0 decreases, as the position of thequantum dot approaches the position of the

1µm

Emission lifetime τ (ns)γ
F≡ γ =

ing reasons: (i) the height of the post wasinsufficient and part of the DBR underneathis remained un-etched when the post wasformed, resulting in increased diffraction
losses of reflected light, and (ii) the shape of
the post has a sharply tapering structure thatcauses transverse radiative losses.
The Purcell factor γ /γ 0 increases as the

0.5

0.4

0.3
894 896 898
Wavelength (nm)Fig. 9: Variation of emission intensity and spontaneous emission lifetime
with emission wavelength
for microcavities with post
diameters of (a) 2 µm and(b) 0.5 µm.

(b) 61
0.8

Q=12700.6

0.7 nm

0.4
0.2
0

918
920
922
Wavelength λ (nm)Spontaneous emission rate
1/ τ (1/ns)

Emission intensity(arbitrary units)

(a)

5
4
3
2
1
0

0.5
1
1.5
2
Detuning | λ QD- λ c | (nm)BosonsFermions

Emission intensity(arbitrary units)(c) 0.8
0.6

τoff =976 ps

0.4
0.2

τon =188 ps

0
0

0.5

1

1.5

2

Time (ns)Fig. 10: (a) Characteristics of the fabricated DBR post microcavity. (b) Purcell factor vs. detuning.
(c) Spontaneous emission processes when the exciton emission wavelength of the quantum dot matches
the cavity resonant wavelength, and when it does not match the resonant wavelength.

maximum electric field, and as the exciton
emission wavelength approaches the cavity
resonant wavelength. A larger Purcell factor
results in a greater reduction ratio of the exciton emission lifetime. Achieving a large Purcell factor γ /γ 0 and a spontaneous emission
coefficient β close to 1 requires that the microcavity has a large Q and small V 0.
But as shown in Fig. 7(a), the post shapeproduced by conventional ECR dry etchinghas a tapering structure caused by undercutting. This causes the radiative losses toincrease and as a result the magnitude of
Q is kept down to 300-800 or thereabouts.
This problem was addressed by switching tochemically assisted ion beam etching (CAIBE),
resulting in the successful fabrication of apost structure with little tapering as shownin Fig. 7(b). This technique was used to produce a GaAs/AlAs DBR post microcavity witha post diameter of 0.4 µ m and a post heightof 5 µ m with suppressed taper-related radiative losses, which was able to exhibit the following characteristics: Q=1300, V0=1.5(λ /n)3,
γ /γ 0=6 and β =0.84 (Fig. 10). It was also possible to improve the external quantum efficiency of photons extracted as a simple Gaussian
beam to 38%.22)

4. Generating a single photon
pulse at the Fourier transform
limit
4.1 Indistinguishable quantum
particles
The lifetime of excitonic emissions froma quantum dot is normally 0.5-1 ns. On theother hand, the decoherence time of exciton
polarization caused by phonon scattering isabout 1 ns at a temperature of 4 K.23) That is,
the natural width of spontaneous emissions
is of about the same order as the homogeneous broadening caused by phonon scattering. If the lifetime of excitonic emissions froma single quantum dot can be reduced using athree-dimensional microcavity, then it should
be possible to reduce the excitonic emission
linewidth to the natural width limit determined by the emission lifetime, and to reduce
the temporal width and spectral width ofsuccessively radiated single photon pulses to
the Fourier transform limit. A single photon
pulse stream of this sort should give rise toquantum interference phenomena as indistinguishable quantum particles.
Figure 11 shows an example of such
quantum interference. When two identicalparticles 1 and 2 are known to have one eachof the two states | r A 〉 and | r B 〉, the system

Fig. 11: Collision characteristics of indistinguishable
identical quantum particles

state is expressed as a completely symmetricor completely antisymmetric wave function.

Ψ 12 〉 =1[ 1, γ A ; 2, γ B 〉2± 2, γ A ; 1, γ B 〉 ]The former applies to bosons, and thelatter applies to fermions. This is called thesymmetrization postulate. When two identical particles are incident on a 50-50% beam
splitter, bosons both appear simultaneously
at the same output port, but fermions arealways output from different ports. This is acharacteristic phenomenon of quantum particles that occurs due to quantum interference
between the direct term and the exchanging
term (the first and second terms respectively in the above equation) of symmetric andantisymmetric wave functions.24) This quantum interference phenomenon is the sourceof the Pauli exclusion principle for fermionsand phenomena exhibited by bosons such as
stimulated emission (lasers), Bose condensation and superconductivity. The scatteringcharacteristics of indistinguishable quantum
particles in a 50-50% beam splitter shown in
Fig. 11 can also be used for the identificationof EPR-Bell states and form the basis of quantum teleportation,5) quantum repeaters,6) andlinear optical quantum computers.7)

JSAP International No.10 (July 2004)

9

Interference strength

Pulse width : 145ps

Intensity

Coherence time : 204 ps
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

4.2 Two-photon interference
experiments

100 200 300 400 500 600 700 800

0

0

0.5

1

1.5

2

Time (ns)

Delay time (ps)Fig. 12: Results of using a Michelson interferometer to measure the coherence time of a single photon
pulse and a streak camera to measure the pulse duration.

t1

t2

(c)
Simultaneous counting probability(a) Photon detector
Beam splitter

2 ns

2 ns

Single mode fiber

Corner cube

Count

1200 (b)
1000
800
600
400
200

2 4
1 3 5

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

-10

-20

-10

0

10

20

30

-400

-200

Delay time τ =t2-t1 (ns)

0

200

400

Delay time (ps)Fig. 13: Collision experiment involving two single photons generated with a 2 ns delay time difference.
When the wave functions of the two single photons were overlapped at the output 50-50% beam splitter,
the simultaneous counting probability was found to be lower due to two-photon interference.

50-50%
beam splitter

Quantum dot
1 2

e

a

b

laser

ˆ1e
ˆ 2 | 0〉
|ψ0 〉= eê11( â1 + b̂1 )
2

|ψ0 〉

1
1
(
+
) + ( â1 â2 + b̂1 b̂2 )2 â1 b̂2 â2 b̂1
2

ê2

1
( â2 + b̂2 )
2

Bob
1 2 3

θ

T

φ

Alice
1 2 3

EPR-Bell stateFig. 14: An entanglement based quantum cryptography protocol using two single photons and a 50-50%
beam splitter

10

JSAP International No.10 (July 2004)

Figure 12 shows the results of using a
Michelson interferometer to measure the coherence time of a single photon pulse stream
generated from this single quantum dot microcavity (coherence time τ c: 200 ps) and the results of using a streak camera to measure thepulse duration (amplitude decay time 2τ rad:290 ps).13) This only differs from the Fouriertransform limit 2τ rad=tc by a factor of 1.5.
Figure 13(a) shows the collision experiment for two single photons generated at
2 ns time intervals. The delay time of theMichelson interferometer was set to 2 ns,
which is the same as the interval between
the two single photon pulses. Figure 13(b)shows the joint counting probability withwhich the two detectors T1 and T2 detected
the photons at t = 0 and t =τ .13) Peaks 1 and
5 in the central cluster centered on t = 0 correspond to the case where the first photon
takes a short path and the next photon takes
a long path. Peaks 2 and 4 correspond tothe case where the two photons both takea short path or both take a long path. The
central peak 3 corresponds to the case where
the first photon takes a long path and thenext photon takes a short path, and only inthis case are the two single photons incident
on the output 50-50% beam splitter simultaneously. The fact that peak 3 is smaller thanpeaks 2 and 4 shows that quantum interference is actually occurring in the identical boson particle collisions shown in Fig. 11. Figure 13(c) shows that this suppression of thesimultaneous counting probability reaches amaximum when the two pulses overlap completely, and disappears when the delay timeexceeds the pulse width 2τ rad : 290 ps. The
dip in simultaneous computation probability caused by two-photon interference shownin Fig. 13 is deeper than the value predictedfrom the offset from the Fourier transform
limit (1.5 times) shown in Fig. 12. This might
be because the coherence time in Fig. 12
does not reflect the actual homogeneous
linewidth but includes the effects of a small
drift in the emission wavelength due to the
inflow and outflow of carriers in carrier traps
near the quantum dot.

the first photon will havetaken a short path anda0 | 0〉c + a1 |1〉cthe next photon will
Classical information (2 bit)
EPR-Bellhave taken a long path,state analysis
in which case there willφ
be a perfect correlation
with respect to the timesof detection, and a se1[ |0〉 b | c + |1〉b | 0〉c ]Beam
2cret key can be derived
splitterfrom the result. Thisa0 | 0〉a + a1 |1〉asort of quantum cryp|1〉b | 0〉c
tography relies on theproperty that two single
Single photon
photons form indistinguishable identical quanFig. 15: Quantum teleportation using a single photon at the Fourier transform limit
tum particles. A slightly
different version of generating an entangled photon-pair from two
the results, and the initial state of mode a is
indistinguishable single photons has been rereproduced in mode c.27) At this time, two26)
cently demonstrated.
single photon pulses that are incident simulFigure 15 shows an example of a contaneously in mode a and mode b must evifiguration for quantum teleportation using adently be indistinguishable identical quantum
single photon pulse at the Fourier transformparticles. This quantum teleportation gate islimit. The single photon pulse is converted ina fundamental building block for linear opto an EPR-Bell state by a 50-50% beam splittics quantum computation 7) and has beenter.
successfully demonstrated in a recent experiment.28)
11〉 b 0〉 c 씮[ 0 〉 b 1〉 c + 1 〉 b 0 〉 c]
2

4.3 Conversion into an entangled
photon pairFigure 14 shows that it is possible toimplement an entanglement based quantum cryptography protocol using two suchsingle-photon pulses at the Fourier transform
limit.25) In this protocol, the two indistinguishable single photons are simultaneously sent
to Alice and Bob respectively by a 50-50%
beam splitter. If Alice and Bob both detect
the photons individually at time 2, then thefirst photon will have taken a long path andthe next photon will have taken a short pathbefore being detected by both parties. However, it is impossible to tell if Alice detected
the first photon 1 ( |1,L 〉A ) and Bob detected
the next photon 2 ( |2,S 〉B ), or vice versa. Thissituation is called a time-energy entangledstate, and is expressed as follows:1iθ
iφ[e 1, L 〉 A 2, S 〉 B +e 2,
2S 〉 A 1, L 〉 B ]Ψ 12 〉 =where θ and φ are the random phasemodulations introduced into Alice and Bob’s
delay lines (0 or π/2). When their two phasesθ and φ are the same, there is a perfect correlation with respect to which detector detected the photon, and it is possible to make asecret key from this result. When their phasesθ and φ are different, there is no correlation
between their detector results, in which casethe measurement results are discarded. When
Alice and Bob’s photon detection times are
1 on the one hand and 3 on the other, thenHere, mode b is obtained by mixing arbitrary incident linear superimposed states
a 0|0〉a +a 1|1〉a with a separate 50-50% beam
splitter, and the output is subjected to Bellstate analysis. Mode c is controlled based on

BB84–Poisson-distributed light source

BB84–Poisson-distributed light source
BB84–Single photon source

BB84–Single photon source
BBM92–Poisson photon pair light source

1×10

BBM92–Single photon pair light source

1×10-3

1×10-5

1×10-7

1×10-9
0

40

80
120
160
Fiber length (km)200

BBM92–Poisson photon pair light source-1

Key creation rate per pulse

Key creation rate per pulse

1×10

Figure 16 shows a theoretical predictionof how the transmission speed of BB84 quantum cryptography using an ordinary semiconductor laser light source and a single photon

0.8 µ m free-space propagation system

1.5 µ m fiber system-1

5. Quantum cryptographyusing single photons

BBM92–Single photon pair light source

1×10-4
1×10-7

1×10-10-13

1×10

1×10-16020

40 60 80 100 120
Repeater gain (dB)

Fig. 16: A comparison of the performance ofquantum cryptography using the BB84 andBBM92 protocols in the 1.5 µm and 0.8 µm
bands. In BB84, the key creation rate perpulse and the repeater gain are substantiallybetter when using a single photon sourcecompared with an ordinary semiconductor laser. In BBM92 a better key creation
rate per pulse is achieved by using a lightsource that generates regular photon pairs
compared with an ordinary parametric lightsource.

JSAP International No.10 (July 2004)11

60 dB in a 0.8 µm band system or increase
the fiber length to 100 km in a 1.5 µm bandsystem. The ultimate secure key creation rate
was recently calculated for cases where asingle photon source emits two or more photons with a finite probability (g(2)(0)≠0) andsource (final key creation rate per pulse after
error correction and privacy amplification)varies with fiber length (1.5 µm band) or repeater gain in the case of spatial propagation(0.8 µm band). When an ordinary semiconductor laser light source is used, there is apossibility of generating two or more photons per pulse due to the Poisson-distributed
photon numbers, allowing an eavesdropper
to use a photon splitting attack. To reduce
this possibility, it is essential to suppress theaverage number of photons per pulse to avalue less than 1. As the fiber length or repeater gain increases, the average number ofphotons per pulse becomes even smaller, andeventually becomes smaller than the average dark count of the detector. At this stage,
the bit error rate becomes very large, making secure key creation impossible. This limit
corresponds to a repeater gain of 20 dB for a
0.8 µm band space propagation system or afiber length of 20 km for a 1.5 µm band fiber
system. However, if an ideal single photon
source is used, the probability for 2 or more
photons per pulse is suppressed to 0, so theaverage number of photons sent out to the
propagation path can be set to 1. This makes
it possible to increase the repeater gain to(a)Counter
PinholeSingle mode
fiber

Cryostatfigure shows, a strong correlation is formed
when Alice and Bob use the same polarization base. After that, classical error correction
is performed using a block coding technique.
To increase the level of security so as to suppress the leakage of information to eavesdroppers to a level where it is ultimately negligible, classical privacy amplification is finallyemployed. Figure 17(c) shows how the repeater gain varies with the final key creation
rate per transmitted pulse in this system. In
the present system, since the average numberof photons sent out to the transmission path
per pulse is 3×10-3, it is inferior to schemes
using an ordinary semiconductor laser in theregion of small repeater gain (propagationloss), but in the high repeater gain region thesmall value of g (2)(0) makes it advantageous
to use a single photon source. In the future
it is expected that the theoretically predicted
repeater gain of 60 dB will be assured by reducing optical losses inside a transmitter.

where the external quantum efficiency is lessthan 1 (η <1), and even in these cases it was
found that the resulting characteristics are
better than when an ordinary Poisson-distributed light source is used.29)
Figure 17(a) shows a BB84 quantum
cryptography test system using the single
photon source developed here.14) This system generates single photons at a repetition
frequency of 76 MHz using a mode-locked
Ti:Al 2O 3 laser. Alice uses random variables
generated in her computer to modulate thepolarization state to one of four states – horizontal (H), vertical (V), right-handed circular(R) or left-handed circular (L) – and sends it
to Bob. Bob then splits the received photons
into two paths with a 50-50% beam splitter and detects one path on an H-V basis andthe other on an R-L basis. Since the photons
are randomly divided between either of theoutput ports, this demodulation scheme iscalled passive demodulation. After this quan-

Alice
Lens

tum transmission has taken place, Alice andBob publicly compare the polarization bases
they used, and store only the data for cases
where they matched. Figure 17(b) shows ahistogram of Alice and Bob’s data.14) As thisPhotodetector

Half-wavelength plateSlit

λ /4 plate

ChannelElectroopticalmodulator

Polarization beam
splitter

Polarization
beam splitter
Lens

Pump light
TIA

Grating
Randomvariable
generator

Amplifier

TIA
Bob

(b)

(c) 104

Key creation rate (bit/s)

0.5
0.4
0.3
0.2
0.1
0

BobAliceBob- L
R AliceBob- R
L AliceTransfer speed 70 kHz
H Alice- Bob- V
Bit error rate 2.5%
H
V

103

Laser scheme (theoretical)
Single photon source scheme (theoretical)
Single photon source scheme (experimental)
Laser scheme (experimental)

102
101
100
10-1
10-2

10

Fig. 17: An experimental BB84 quantum key distribution system using a single photon source, and the results obtainedwith this system

12

JSAP International No.10 (July 2004)

15

20

Repeater gain (dB)2530

Biexciton state(2e- +2h+ )σ+

σ-Real part

Imaginary partσ-

Crystal ground state

0.2
0

1
- +
+
(| σ 1σ 2 〉 + | σ 1, σ 2 〉)2
= 1 (|H1H2 〉+ |V1V2 〉)
2

HH HV

VV

ρ=

1
[ σ + 〉 11 〈 σ + ⊗ σ – 〉 22 〈σ –2+ σ – 〉 11 〈 σ – ⊗ σ + 〉 22 〈σ + ]Figure 19 shows the results of experiments performed to confirm this fact. 31)
Measurements were made of all the ele-

(a)

0.2
0
HH

HV

VH

VV

HH
HV
VH
VV

ments in a 4 × 4 polarization matrix representing the polarization states of biexcitonic
photons and excitonic photons. As a result,it was found that the polarization state of
the two photons is not the expected entangled state but a mixed state consistingof |H〉1|H〉2 and |V〉1|V〉2. The reason for this isshown in Fig. 20.31) When light emitted fromthe quantum dot is separated into H-polarized and V-polarized components which
are input to a Michelson interferometer andthe interference fringe is measured, they areeach found to be exponentially decaying.
The H-polarized and V-polarized waves havedifferent decay constants because the resonant wavelength of the cavity is anisotropic1[ σ + 〉 1 σ – 〉 2+ σ – 〉 1 σ + 〉 2]
2

If the emission time interval between thefirst and second photons becomes longer
than the spin phase relaxation time, then any
information about which route was taken by
the system is lost to the external bath, so thestates of the two photons form a mixed state
as follows:

Interference fringe intensity

Figure 18 shows the two photon emission process when two electrons and twoholes are confined at the ground state ina single InAs quantum dot. According to
the anti-symmetrization postulate of quantum mechanics, the two electrons and holes
should be in respective spin singlet states.
These two electron-hole pairs release
two photons, but are subject to the so-called
selection rules whereby an up-spin electron(J = +1/2) only recombines with an up-spinhole (J = +3/2) to release a right-handed circular polarized (σ +) photon, and a downspin electron (J = -1/2) only recombines witha down-spin hole (J = -3/2) to release a lefthanded circular polarized (σ -) photon. The
first biexcitonic photon to be released has
equal probabilities (50%-50%) of being σ +or σ -, while the second excitonic photon to
be released should have the opposite polarization. If the time interval between the release of the first and second photons is much
shorter than the spin phase relaxation time ofelectrons and holes, and also if the energies
of spin-up and spin-down excitons in intermediate states become degenerate, then it is
impossible to tell which route was taken by
the system and the states of the two photons
enter the following polarization triplet entangled state:30)

Ψ 〉=

VH

HH
HV
VH
VV

Fig. 19: Measurement results of a 4×4 matrix representing the polarization states of biexcitonic
photons and excitonic photons from a quantum dot. The two photons form the mixed states|H〉1|H〉2 and |V〉1|V〉2.

Fig. 18: An illustration showing how biexcitonic photons and excitonic photons from a single quantum dot are in a polarization entangled state

6. Direct generation of
EPR-Bell photon pairs

0.6
0.4

0.6
0.4

(b)

1
Horizontally polarized

0.5
0
1

Unpolarized

0.5
0
1

Vertically polarized

0.5
0
-200

0

200

400

600

1 delay time (ps)Horizontally
polarized

Emission intensityσ+0.55 nm
931.8 nm

Verticallypolarized

1 nm
931.2 nm
932 nmQuantum dot

929

930

931

932

933

934

Wavelength λ (nm)Fig. 20: (a) Variation of interference fringe intensity with delay time for H-polarized, V-polarized and unpolarized light.
(b) Cavity resonance characteristics and quantum dot excitonic emission spectra for H-polarized and V-polarized waves.

JSAP International No.10 (July 2004)

13and the exciton emission wavelength is close
to the resonant wavelength for H-polarizedwaves. If the H-polarized and V-polarizedwaves are introduced into the Michelson interferometer without separating them first,
then the interference fringe is found to oscillate with a period of 200 ps as shown in Fig.
20(a). The period of this oscillation showsthat the energy difference between the H-polarized and V-polarized photons is 13 µ eV.
That is, the energy of the excitons in the two
spin states comprising the intermediate state
are not degenerate. This is thought to be due
to such effects as residual stress and anisotropy of the quantum dot shape. Thus the system ends up in the mixed states |H〉1|H〉2 and|V〉1|V〉2 because the information about whichpaths the two photons were released along
leaks out from the system via the energy ofthe excitonic photon.
If a microcavity can be used to make theexciton emission lifetime much shorter than
200 ps, then the energy difference between
the H-polarized and V-polarized waves would
be hidden by natural linewidth. It therefore
remains possible that polarization entangledstates can be generated directly.

7. Prospects
The generation of a regular single photon
stream at the Fourier transform limit is a new
development in quantum information processing using photonic qubits. This technique
has been used in various quantum information systems, including the generation of entangled photon pairs, BBM92 quantum cryptography, quantum teleportation, quantum
repeaters and linear optical quantum gates.
When considering the future quantum
information technology, it will be necessary to
encode information on the photonic qubits inquantum communication but store information on the nuclear spin qubits in quantum
memory. Electron spins will no doubt play animportant role as an interface between the
two. It is hoped that a great deal of wisdom
will be developed on how to connect thesethree qubit systems.

14

JSAP International No.10 (July 2004)

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Yoshihisa Yamamoto
Received a Ph.D. from the University of Tokyo in 1978, and has been working at Stanford University as a Professor of Applied Physics and Electrical Engineering since 1992 and at National Institute of Informatics as a Professor since 2003.
He is also an NTT R&D Fellow, and a representative of the JST international joint
research project. His current research interests include quantum optics, mesoscopic
physics, semiconductor nanostructures, solid-state NMR spectroscopy and quantum information.